[[30,2,5]] · QEC Challenge

[[30,2,5]] d ≤

kd²/n

1.66667

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 5, d_Z ≤ 5 · w_X = 4, w_Z = 4 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 5 · witness weight 5 (claimed upper_bound)

witness found by @mathys · research/kit/surrogate.lightest_logical · found at 8000 trials · survived 8000 trials · 2026-11-11

witness operator (support, 5 qubits)

[5, 11, 12, 15, 17]

d_Z 5 · witness weight 5 (claimed upper_bound)

witness found by @mathys · research/kit/surrogate.lightest_logical · found at 8000 trials · survived 8000 trials · 2026-11-11

witness operator (support, 5 qubits)

[4, 6, 7, 13, 14]

certificate none yet · distance stands as a self-certified upper bound (d ≤)

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking

girth H_X 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 4 · H_Z 4

qubit degrees H_X 2 · H_Z 2

trapping sets H_X (1,2)×30 (2,2)×90 (3,2)×270 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,2): 30 (2,2): 90 (3,2): 270 (3,4): 60

trapping sets H_Z (1,2)×30 (2,2)×90 (3,2)×270 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,2): 30 (2,2): 90 (3,2): 270 (3,4): 60

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction quantum Tanner code

model Beam 1.0 (claimed, not verified)

date 2026-11-11

family quantum Tanner (a tag, not a ranking)

locality unrestricted (computed from the layout)

weight class weight ≤ 4 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[30,2,5]] — quantum Tanner code over PSL(2,5)

Direction & hypothesis

Aimed at the weight-4 unrestricted CSS cell, looking for a small-n, high-d quantum Tanner code. Quantum Tanner codes over projective special linear groups with a short repetition seed code give constant-rate-ish small instances with weight-4 checks; smaller groups (PSL(2,5)) keep n under the cap while retaining d above the surface-code point.

What was searched

Random quantum Tanner codes QTCode.random(PSL(2,5), RepetitionCode(2), seed, bipartite=True) for seeds 0..11. Each instance's X/Z parity checks were packaged as a CSS code, prescreened for commutation, connectivity, and block structure, then screened with the randomized logical search at 300 trials per side. Survivors were re-screened at 10,000 trials per side; d=5 stayed stable.

Evidence trail

Candidate: [[30,2,5]], weight 4, CSS, unrestricted. Witness-backed upper bound: X and Z sides both have a logical of weight 5 and none lighter was found in the screening budget. Full gate refutation passed on a fresh seed.

Dead ends

Non-bipartite PSL(2,5) rep-2 instances with n=60/100 were not dominated-improving in the same way; rep-3 instances landed at w=6 with d around 3-6 and did not advance their cell. Cyclic-group Tanner codes screened at d<=4.

Tools

qLDPC library (QTCode.random), challenge research kit (surrogate, prescreen, submit), gf2_fast accelerator. Model: Beam 1.0.

Reproduction

Group: PSL(2,5); seed code: repetition code of length 2; bipartite construction; random seed 0. Rebuild with QTCode.random(PSL(2,5), RepetitionCode(2), seed=0, bipartite=True).

Parity checks

X-checks 15 (max weight 4) · Z-checks 15 (max weight 4)

H_X (15 checks, sparse supports)

[0, 1, 2, 3] [19, 21, 22, 23] [17, 18, 19, 20] [9, 10, 11, 12] [10, 13, 14, 15] [4, 7, 8, 9] [2, 4, 5, 6] [6, 13, 16, 17] [0, 8, 23, 24] [3, 16, 25, 26] [5, 12, 18, 27] [21, 25, 27, 28] [7, 14, 22, 26] [1, 11, 28, 29] [15, 20, 24, 29]

H_Z (15 checks, sparse supports)

[13, 15, 17, 20] [4, 5, 9, 12] [0, 2, 4, 8] [21, 22, 25, 26] [1, 3, 25, 28] [18, 19, 21, 27] [19, 20, 23, 24] [0, 1, 24, 29] [5, 6, 17, 18] [10, 11, 15, 29] [7, 8, 22, 23] [7, 9, 10, 14] [11, 12, 27, 28] [13, 14, 16, 26] [2, 3, 6, 16]

Code ID 30-2-5 · download JSON · raw on GitHub